There's an interesting discussion raging on elsewhere in the thread about the possibility for cultural adaptation that would enable us to cope with statements like "the cancer test was positive, but there's still a decent chance you don't have cancer".

There's also an interesting question, taking your roughly 1:1 true to false positive ratio as an example, of whether the marginal true positive does more good for the world than the marginal false positive.

Side-stepping those questions though, they probably won't give this test to literally every person in the population. How much does the picture improve when you give this only to people over 50? Or only smokers? You could still massively increase effective screening for cancer with a cheap and easy test if you combine it with enough population filtering to increase the true positive/false positive ratio to something like 70-90% rather than the 50%